Sixty-two per cent of Iraqis voted in yesterday's general election, with both the conservative incumbent prime minister, Nouri al-Maliki, and his secular rival, Iyad Allawi, claiming today to have performed strongly in the poll.

The turnout was lower than the reported 76% turnout in the 2005 poll, which was boycotted en masse by two provinces, but higher than in last year's provincial ballot, the electoral commission said.

The lower nationwide tally this time seems to have been balanced by a higher than 50% turnout across all provinces, including the restive Sunni heartland. Preliminary results in the landmark poll are expected by Thursday, and several months of horse-trading to select a prime minister are expected to follow.

The pre-poll frontrunners, Maliki and Allawi, were both talking up their performances, although neither man is likely to be able to form a government without entering into a coalition with at least one other rival.

Officials in Maliki's heartland in the Shia south of Iraq said his political list had won in at least three provinces. Allawi, a secular Shia Muslim, is believed to have polled well in Baghdad and, perhaps surprisingly, in the exclusively Sunni Anbar province, as well as Sunni areas of Baghdad. He needs his cross-sectarian platform to take hold across large sections of Iraqi society to stand a hope of being elected for a second time as leader.

There are no indications yet of widespread voter fraud and US officials in Iraq today cautiously predicted that the mass withdrawal of their forces would begin in earnest about 60 days after a final result was determined.

The US commanding general, Ray Odierno, said: "Today I believe we are going to be on 50,000 [troops] by 1 September. We believe we are right on track."

That number matches a pre-election pledge to withdraw all combat troops from Iraq by the end of the summer as part of a security agreement signed between Baghdad and Washington. Odierno said 35,000 US troops had left Iraq since September and another 45,000 were expected to pull out between May to November.

He said the barrage of explosions that reverberated around Baghdad as voting got under way yesterday were caused by dozens of bottles packed with explosives and detonated by mobile phones, not mortars and rockets, as the Iraqi army had previously claimed.

The devices seemed aimed more at deterring voting than at causing widespread carnage, although up to 35 people were killed and several dozen more wounded, according to two Baghdad hospitals visited by the Guardian. At least two buildings were rigged with explosives and blown up.

"We did not have a single track yesterday," Odierno said, in reference to radar sites that track the launch and trajectory of the rockets and mortars that regularly pepper Baghdad's international zone and some US bases.

He said joint Iraqi and US forces had intercepted 20 suicide vests in the three days before the election as well as three to four vehicle bombs and a cell of would-be female suicide bombers in Diyala province, north of the capital.